Section3.3Accumulation Sequences

Overview

One of the most important mathematical ideas in calculus is that of an accumulation of change for physical quantities. As we have been learning about sequences, we have talked about how we can define sequences using explicit formulas and using recursive definitions. More recently, we have looked at how the increments of a sequence can help us understand the behavior of a sequence. For some sequences, we learned that patterns in the increments could be used to find additional terms in a sequence.

We are now ready to think about this more generally. Given any sequence of values, we wish to find that sequence for which the given sequence matches the increments. We call the sequence that we are finding the accumulation sequence of the given sequence.

In this section, we formally define and discuss the theory of accumulation sequences. Summation notation is introduced. We establish conditions that guarantee two sequences are equivalent. Then we illustrate applying these conditions to demonstrate that the explicit and recursive definitions for arithmetic and geometric sequences are equivalent.

Subsection3.3.1Accumulation of Change

There are many examples of quantities where we track changes to the quantity rather than repeated measure the quantity itself. Consider a bank balance. We do not count our money every month. Instead, we add up all of our deposits and withdrawals and use them to adjust our record for the balance. Similarly, consider a population under study. It could be very costly to count all of the individuals every month. If instead we could track how many births and deaths occurred during the month, we could calculate a new population count by adding births and subtracting deaths.

Example3.3.1

At the start of the year, you had $1500 in an account. Suppose that the sequence

Let \(B\) represent the monthly balance. Before any months pass, we have a balance of 1500 dollars. This gives an initial value \(B_0 = 1500\text{.}\) We wish to define the sequence \(B= (B_m)_{m=0}^{12}\text{.}\)

After one month, our account has had $240 withdrawn and $280 deposited. The balance after the end of the month is thus given by

Once we have the balance after one month, we can repeat this process for the other eleven months.

\(m\) (month)

\(B_m\) (balance in dollars)

0

1500

1

\(1500-240+280 = 1540\)

2

\(1540-300+280 = 1520\)

3

\(1520-270+280 = 1530\)

4

\(1530-450+280 = 1360\)

5

\(1360-250+280 = 1390\)

6

\(1390-310+280 = 1360\)

7

\(1360-360+280 = 1280\)

8

\(1280-270+280 = 1290\)

9

\(1290-320+280 = 1250\)

10

\(1250-300+280 = 1230\)

11

\(1250-350+280 = 1160\)

12

\(1160-480+280 = 960\)

When we create a sequence of values based on knowing the increments, we are creating what we call an accumulation sequence.

Definition3.3.2

Given a sequence \(x = (x_k)_{k=m}^{n}\text{,}\) we say \(u\) is an accumulation sequence of \(x\) if \(u = (u_k)_{k=m-1}^{n}\) with \(\nabla u_k = x_k\text{.}\)

Subsection3.3.2Equivalent Sequences

A given sequence of increments has infinitely many different accumulation sequences which differ in their initial value. However, for a given initial value and sequence of increments, the resulting accumulation sequence is unique. That is, any two sequences that have the same initial value and increments sequences that are equal for all values, then the sequences themselves are equal for all values.

Theorem3.3.3Uniqueness Conditions for Accumulation Sequences

Given two sequences \(u\) and \(w\text{.}\) If \(u_m = w_m\) and \(\nabla u_k = \nabla w_k\) for all \(k \gt m\text{,}\) then \(u_k = w_k\) for all \(k \ge m\text{.}\)

In mathematics, to prove that every statement from a sequence of statements is true, we often use an approach called the Principle of Mathematical Induction. This requires demonstrating that the first statement in the sequence is true, and then showing that anytime one of the statements is true, the subsequent statement must also be true.

This theorem is perfectly suited to apply mathematical induction. The sequence of statements we wish to prove is

The first statement in the sequence, \(u_m = w_m\) is true by assumption—one condition is that the sequences \(u\) and \(w\) have the same initial values. The inductive step is to go from an arbitrary statement in the sequence of statements to the next. So suppose \(u_k = w_k\) for some index \(k\) in \(\{m,m+1,\ldots\}\text{.}\) We know that \(\nabla u_{k+1} = \nabla w_{k+1}\) by the assumption that the sequences have equal increments. We now use substitution twice:

The sequences \(x\) and \(y\) have the same initial value and the same increments. Therefore, they have all the same values: \(x_k = y_k\) for all \(k=1, 2, \ldots\text{.}\)

Theorem Theorem 3.3.3 can be generalized from having two sequences with equal increments to two sequences sharing any recurrence relation involving the previous term. For example, a geometric sequence has a recurrence relation \(x_{n} = \rho x_{n-1}\text{,}\) so that the increment using the relation itself depends on the previous term, \(\nabla x_n = (\rho - 1)x_{n-1}\text{.}\)

Theorem3.3.5

Suppose \(u\) and \(w\) are two sequences with common initial values, \(u_m = w_m\text{.}\) If there is a sequence of projection functions \(f_k\) so that \(u\) and \(w\) satisfy the same relations,

\begin{equation*}
u_{k} = f_k(u_{k-1})
\end{equation*}

and

\begin{equation*}
w_{k} = f_k(w_{k-1})\text{,}
\end{equation*}

then \(u_k = w_k\) for all \(k=m,m+1,\ldots\text{.}\)

For a recursively defined sequence, the sequence of projection functions would all be the same function.

We next compare the recurrence relations. We know that \(y\) has projection function \(f: y_{n-1} \mapsto y_n = \frac{1}{2}y_{n-1}\text{.}\) We need to show that \(x\) satisfies the same relation, \(x_k = \frac{1}{2} x_{k-1}\text{.}\) Using the explicit formula for \(x\text{,}\) we compute both sides of the recurrence equation and show they are equivalent.

Comparing the formulas, we see that \(x_k = \frac{1}{2} x_{k-1}\text{.}\)

The sequences \(x\) and \(y\) have the same initial value and the same sequence of recurrence relations. Therefore, they have all the same values: \(x_k = y_k\) for all \(k=1, 2, \ldots\text{.}\)

We end our discussion of showing two sequences are equivalent by establishing an explicit formula for sequences defined recursively by a linear projection function,

\begin{equation*}
x_{n} = \alpha x_{n-1} + c\text{,}
\end{equation*}

with \(\alpha \ne 1\text{.}\) When \(\alpha = 1\) we have an arithmetic sequence, which is a sequence we already know. When \(c=0\text{,}\) we have a geometric sequence. The projection function \(f:x_{n-1} \mapsto x_{n}\) is defined by the formula \(f(x) = \alpha x + c\text{.}\) The fixed point \(x^*\) is the solution to

where \(\displaystyle x^* = \frac{c}{1-\alpha}\) is the equilibrium of the sequence.

Subsection3.3.3Summation Notation

In mathematics, the idea of adding terms from a sequence appears so frequently that a special notation, called summation notation or sigma notation for the Greek letter sigma \(\Sigma\text{,}\) was created to represent the sum.

Definition3.3.8Summation Notation

Given any sequence \(x\) and integers \(m \le n\text{,}\) the sum of terms \(x_k\) with index \(k\) satisfying \(m \le k \le n\) is written

An accumulation sequence is closely related to summation. The accumulation sequence is a new sequence formed by starting with an initial value and then adding one increment at a time. Suppose \(x=(x_k)_{k=1}^{\infty}\) and \(u\) is the corresponding accumulation sequence with initial value \(u_0\text{.}\) We can write each term of \(u\) as the initial value plus a partial sum of the increments.

Notice how the index for \(u\) appears as the upper limit of the summation and that the index of summation is a different variable. The index of summation can be any other unused variable, so that we might have instead written

We need an explicit formula for the sequence \(k \mapsto a_k\text{.}\) We recognize that \(a\) is an arithmetic sequence with \(a_1 = 3\) and constant increment \(\nabla a_k = 2\text{.}\) By Theorem 2.2.8, we know \(a_k = a_1 + 2(k-1) = 2k + 1\text{.}\) Using this explicit formula in the summation, we find

The explicit formula for the increment of \(w\) is the same as that for \(u\text{.}\) Consequently, we know that \(u_n = w_n\) for all \(n=0, 1, 2, \ldots\text{.}\)

Subsection3.3.4Summary

An accumulation sequence is a sequence generated from an initial value and a given sequence of increments.

If \(x\) is the sequence of increments and \(u\) is the accumulation sequence, then \(u\) satisfies the recurrence relation

\begin{equation*}
u_{n} = u_{n-1} + x_n\text{.}
\end{equation*}

If two sequences share the same initial value and the same increments, then the sequences are identical (Theorem 3.3.3). More generally, if two sequences share the same initial value and sequence of recurrence relations involving the previous term, then the sequences are identical (Theorem 3.3.5).

Summation notation (sigma notation) provides a method to communicate the sequence of increments as well as the range of index values. The index variable is sometimes called the dummy variable because any other variable could be used in its place.

Every accumulation sequence \(u\) can be represented as the initial value added to the summation of its increments \(x_k\) with the index variable appearing as the upper limit,

10

11

For each of the following summations, write down the sum of individual terms. Then compute the value of the sum. For example, \(\sum_{k=2}^{5} 2k\) would be \(2(2)+2(3)+2(4)+2(5)=4+6+8+10 = 28\text{.}\)

12

\(\displaystyle \sum_{k=12}^{15} 3k\)

13

\(\displaystyle \sum_{k=-2}^{2} 2^k\)

14

\(\displaystyle \sum_{k=2}^{5} \frac{2k+1}{5k}\)

Rewrite the following sums in summation notation. Find an appropriate formula for the increment sequence and identify the correct lower and upper limits of the sum.